pinky asked in Science & MathematicsMathematics · 3 months ago

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• 3 months ago

Since 4 is a zero, f(x) is divisible by x - 4.

.................3x^2-19x +6

.....--------------------------

x-4|3x^3-31x^2+82x-24

......3x^3-12x^2

......---------------

..............-19x^2+82x

..............-19x^2+76x

...............--------------

.............................6x-24

.............................6x-24

.............................-------

...................................0

3x^3 - 31x^2 + 8x - 24 = 0

(x - 4)(3x^2 - 19x + 6) = 0

(x - 4)(3x - 1)(x - 6) = 0

x = 4 or x = 1/3 or x = 6

• 3 months ago

x = 4 is a root, so x - 4 = 0 is a factor

(x - 4) * (ax^2 + bx + c) = 3x^3 - 31x^2 + 82x - 24

ax^3 + bx^2 + cx - 4ax^2 - 4bx - 4c = 3x^3 - 31x^2 + 82x - 24

ax^3 = 3x^3

a = 3

bx^2 - 4ax^2 = -31x^2

b - 4a = -31

b - 12 = -31

b = -19

cx - 4bx = 82

c - 4b = 82

c + 76 = 82

c = 6

-4c = -24

-24 = -24

(x - 4) * (3x^2 - 19x + 6) = 3x^3 - 31x^2 + 82x - 24

3x^2 - 19x + 6 = 0

x = (19 +/- sqrt(361 - 72)) / 6

x = (19 +/- sqrt(289)) / 6

x = (19 +/- 17) / 6

x = 2/6 , 36/6

x = 1/3 , 6